1. S ystems Analysis Laboratory Helsinki University of Technology Game Optimal Support Time of a Medium Range Air-to-Air Missile Janne Karelahti, Kai Virtanen, and Tuomas Raivio Systems Analysis Laboratory Helsinki University of Technology

2. S ystems Analysis Laboratory Helsinki University of TechnologyContents Problem setup Support time game Modeling the probabilities related to the payoffs Numerical example Real time solution of the support time game Conclusions

3. S ystems Analysis Laboratory Helsinki University of Technology Problem setup One-on-one air combat with missiles Phases of a medium range air-to-air missile: 1.Target position downloaded from the launching a/c 2.In blind mode target position is extrapolated 3.Target position acquired with the missile’s own radar In phase 1 (support phase), the launching a/c must keep the target within its radar’s gimbal limit Prolonging the support phase −Shortens phase 2, which increases the probability of hit −Degrades the possibilities to evade the missile possibly fired by the target

4. S ystems Analysis Laboratory Helsinki University of Technology Problem setup The problem: optimal support times t B, t R ? Phase 1: support Phase 2: extrapolation Phase 3: locked

5. S ystems Analysis Laboratory Helsinki University of Technology Modeling aspects Aircraft & Missiles −3DOF point-mass models −Parameters describe identical generic fighter aircraft and missiles −Missile guided by Proportional Navigation −Assumptions −Simultaneous launch of the missiles −Constant lock-on range −Target extrapolation is linear −Missile detected only when it locks on to the target −State measurements are accurate −Predefined support maneuver of the launcher keeps the target within the gimbal limit

6. S ystems Analysis Laboratory Helsinki University of Technology Support time game Gives game optimal support times t B and t R as its solution The payoff of the game  probabilities of survival and hit The probabilities are combined as a single payoff with weights The weights, i=B,R reflect the players’ risk attitudes Blue’s probability of survivalBlue missile’s probability of hit Blue: Red:  Blue missile’s probability of guidance Blue missile’s probability of reach Blue missile’s prob. of hit =

7. S ystems Analysis Laboratory Helsinki University of Technology Modeling the probabilities p r and p g Probability of reach p r : −Depends strongly on the closing velocity of the missile −The worst closing velocity corresponding to different support times  a set of optimal control problems for both players Probability of guidance p g : −Depends, i.a., on the launch range, radar cross section of the target, closing velocity, and tracking error

8. S ystems Analysis Laboratory Helsinki University of Technology p r and p g in this study predetermined support maneuver optimize: minimize closing velocity extrapolate Probability of reach closing velocity at distance d f Probability of guidance tracking error at

9. S ystems Analysis Laboratory Helsinki University of Technology Minimum closing velocities For each (t B,t R ), the minimum closing velocity of the missile against the a/c at a given final distance d f (here for Blue aircraft): u = Blue a/c’s controls, x = states of Blue a/c and Red missile, f = state equations, g = constraints Initial state = vehicles’ states at the end of Blue’s support phase Direct multiple shooting solution method => time discretization and nonlinear programming

10. S ystems Analysis Laboratory Helsinki University of Technology Solution of the support time game Reaction curve: −Player’s optimal reactions to the adversary’s support times Solution = Nash equilibrium −Best response iteration Red player: Blue player: w B =0 Support time of Blue Support time of Red

11. S ystems Analysis Laboratory Helsinki University of Technology support phase extrapolation phase locked phase x range, km altitude, km y range, km Example trajectories Red (left), w R =0.5, supports 12.4 seconds Blue (right), w B =1.0, supports 5.0 seconds

12. S ystems Analysis Laboratory Helsinki University of Technology Off-line: −Solve the closing velocities and tracking errors for a grid of initial states In real time: −Interpolate CV’s and TE’s for a given intermediate initial state −Apply best response iteration Red: Blue: Support time of Blue Support time of Red interpolated optimized Real time solution

13. S ystems Analysis Laboratory Helsinki University of TechnologyConclusions The support time game formulation −Seemingly among the first attempts to determine optimal support times AI and differential game solutions: the best support times based on predefined decision heuristics Discrete-time air combat simulation models: predefined support times Pure differential game formulations are practically intractable Utilization aspects −Real time solution scheme could be utilized in, e.g., Guidance model of an air combat simulator Pilot advisory system Unmanned aerial vehicles